Question 2 AP STATISTICS 2008 SCORING GUIDELINES (Form B)

(1)

AP

^®

STATISTICS

2008 SCORING GUIDELINES (Form B)

Question 2 Intent of Question

The primary goals of this question were to assess a student’s ability to (1) recognize an unbiased estimator and explain why the estimator is unbiased and (2) compare two estimators with respect to center and variability.

Solution Part (a):

Statistics A, C, and D appear to be unbiased. This is indicated by the fact that the mean of the estimated sampling distribution for each of these statistics is about 75, the value of the true population parameter.

Part (b):

Statistic A would be a better choice because it appears to be unbiased. Although the variability of the two estimated sampling distributions is similar, statistic A would produce estimates that tend to be closer to the true population parameter value of 75 than would statistic B.

Part (c):

Statistic C would be a better choice because it has smaller variability. Although both statistic C and statistic D appear to be unbiased, statistic C would produce estimates that tend to be closer to the true population parameter value of 75 than would statistic D.

Scoring

Parts (a), (b), and (c) are each scored as essentially correct (E), partially correct (P), or incorrect (I).

Part (a) is scored as follows:

Essentially correct (E) if the response contains the following two components.

Component 1: Identifies statistics A, C, and D as the unbiased estimators.

Component 2: Clearly demonstrates an understanding of the meaning of the term unbiased. That is, states that the mean (or center) of each distribution is about 75. No other characteristic (e.g., shape, spread) should be mentioned in the response unless it clearly is discounted as a criterion for being unbiased.

Partially correct (P) if the response contains just one of these components. That is, the response either identifies statistics A, C, and D as the unbiased estimators but gives a weak or no explanation or includes some discussion of another characteristic (e.g., shape, spread) as of some importance in judging bias OR demonstrates clear understanding of the meaning of the term unbiased but identifies only one or two of statistics A, C, and D as unbiased estimators.

Incorrect (I) otherwise.

(2)

AP

^®

STATISTICS

2008 SCORING GUIDELINES (Form B)

Visit the College Board on the Web: www.collegeboard.com.

Question 2 (continued) Part (b) is scored as follows:

Essentially correct (E) if the response gives a clear explanation that statistic A is the better choice because it is centered at 75. (It is not necessary that the term unbiased be used as long as the response clearly addresses the center. It is not necessary to mention that statistics A and B both have about the same variability.)

Note: Students who were penalized in part (a) for mentioning other characteristics should not be repenalized in part (b).

Partially correct (P) if the response shows an understanding that centering is an important issue in determining which statistic is better but either does not apply the concept correctly OR communicates the explanation poorly.

Incorrect (I) otherwise.

Part (c) is scored as follows:

Essentially correct (E) if the response gives a clear explanation that statistic C is the better choice based on the fact that it has less variability. (It is not necessary to mention that statistics A and B are both centered at the parameter.)

Partially correct (P) if the response shows an understanding that variability is an important issue in determining which statistic is better but either does not apply the concept correctly OR communicates the explanation poorly.

Incorrect (I) otherwise.

4 Complete Response

All three parts essentially correct 3 Substantial Response

Two parts essentially correct and the other part partially correct OR

Part (a) and one other part essentially correct and the remaining part incorrect 2 Developing Response

Part (a) essentially correct and one of the other parts partially correct OR

All three parts partially correct OR

Part (a) partially correct and one of parts (b) and (c) essentially correct

(3)

AP

^®

STATISTICS

2008 SCORING GUIDELINES (Form B)

Question 2 (continued) 1 Minimal Response

One part essentially correct and the other parts incorrect OR

Two parts partially correct and the other part incorrect OR

Part (a) incorrect, one of the other parts essentially correct, and one part partially correct

(4)

(5)

(6)

(7)

AP

^®

STATISTICS

2008 SCORING COMMENTARY (Form B)

Question 2

Sample: 2A Score: 4

In part (a) this strong response correctly defines an unbiased estimator as one in which “the mean of its sampling distribution must equal the [population] parameter” and then explains why statistics A, C, and D apparently are unbiased estimators. Part (a) was scored as essentially correct. For parts (b) and (c), the student defines a good estimator as being “both unbiased and with low variability.” Thus, although both statistics have about the same variability, statistic A is the better choice in part (b) because the mean of its estimated sampling distribution is closer to 75. Part (b) was scored as essentially correct. Although the estimated sampling distribution of both statistics have a mean of about 75, the student indicates that statistic C is the better choice in part (c) because its distribution has less variability. Part (c) was scored as essentially correct. The whole answer, with all three parts correct, was judged a complete response.

Sample: 2B Score: 3

In part (a) this response gives the correct reason for selecting statistics A, C, and D as being the apparently unbiased estimators. Part (a) was scored as essentially correct. The choice of statistic A in part (b) is right, with an acceptable explanation, so part (b) was scored as essentially correct. The choice of statistic C in part (c) is also correct, but the reason for this choice is not clear. The two distributions are described separately, but no

comparison is made between the two. Part (c) was scored as partially correct. The overall answer was deemed a substantial response.

Sample: 2C Score: 2

This clearly written response does not reflect an understanding of the definition of an unbiased estimator. It includes the correct criterion in part (a) but also incorrectly implies in the explanation and by selecting only statistic A that normality is a second necessary requirement. Parts (b) and (c) are well done. Because part (a) was worth as much as parts (b) and (c) combined, this answer was considered a developing response.